Mathematics — CBSE Class 10 board question

For equal and real roots, discriminant $D = 0$.

Here, $A = (1+a^2)$, $B = 2ab$, $C = (b^2 - c^2)$.

$$D = B^2 - 4AC = 0$$

$$(2ab)^2 - 4(1+a^2)(b^2 - c^2) = 0$$

$$4a^2b^2 - 4(b^2 - c^2 + a^2b^2 - a^2c^2) = 0$$

$$4a^2b^2 - 4b^2 + 4c^2 - 4a^2b^2 + 4a^2c^2 = 0$$

$$-4b^2 + 4c^2 + 4a^2c^2 = 0$$

$$4c^2(1 + a^2) = 4b^2$$

$$\boxed{b^2 = c^2(1+a^2)}$$

Hence proved.

Source: Chapter 4, Section 4.4 – Nature of Roots

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• The key condition for equal real roots is discriminant = 0 ($b^2 - 4ac = 0$).
• Carefully identify $A$, $B$, $C$ from the given equation (don't confuse with the $a, b, c$ in the equation itself — use different letters or be explicit).
• Expand neatly and cancel $4a^2b^2$ terms — that's where students usually lose marks.
• Always write "Hence proved" at the end for proof-type questions.